Advertisement

Communications in Mathematical Physics

, Volume 244, Issue 2, pp 297–309 | Cite as

Cantor Spectrum for the Almost Mathieu Operator

  • Joaquim PuigEmail author
Article

Abstract

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \({{ {{\left({{H_{{b,\phi}} x}}\right)}}_n= x_{{n+1}} +x_{{n-1}} + b \cos{{\left({{2 \pi n \omega + \phi}}\right)}}x_n }}\) on l 2(ℤ) and its associated eigenvalue equation to deduce that for b≠0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ‘‘Ten Martini Problem’’ for these values of b and ω. Moreover, we prove that for |b|≠0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

Keywords

Real Line Eigenvalue Equation Mathieu Operator Cantor Spectrum Cantor Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnol’d, V.I.: Geometrical methods in the theory of ordinary differential equations. Vol. 250 of Grundlehren der Mathematischen Wissenschaften. New York: Springer-Verlag, 1983Google Scholar
  2. 2.
    Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Preprint, 2003Google Scholar
  3. 3.
    Avron, J., Simon B.: Almost periodic Schrödinger operators II. The integrated density of states. Duke Math. J. 50, 369–391 (1983)zbMATHGoogle Scholar
  4. 4.
    Azbel, M.Ya.: Energy spectrum of a conduction electron in a magnetic field. Soviet Phys. JETP. 19, 634–645 (1964)Google Scholar
  5. 5.
    Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3), 408–419 (1982)zbMATHGoogle Scholar
  6. 6.
    Broer, H.W., Puig, J., Simó, C.: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. Commun. Math. Phys. 241 (2-3), 467–503 (2003)Google Scholar
  7. 7.
    Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. The Probability and its Applications. Basel-Boston: Birkhäuser, 1990Google Scholar
  8. 8.
    Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99(2), 225–246 (1990)zbMATHGoogle Scholar
  9. 9.
    Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955Google Scholar
  10. 10.
    Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, Vol. 629, Berlin: Springer-Verlag, 1978Google Scholar
  11. 11.
    DeConcini, C., Johnson R.A.: The algebraic-geometric AKNS potentials. Ergodic Theory Dynam. Syst. 7(1), 1–24 (1987)Google Scholar
  12. 12.
    Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. 89(3), 415–426 (1983)zbMATHGoogle Scholar
  13. 13.
    Dinaburg, E.I., Sinai, Y.G.: The one-dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975)zbMATHGoogle Scholar
  14. 14.
    Eliasson, L.H.: One-dimensional quasi-periodic Schrödinger operators – dynamical systems and spectral theory. In: European Congress of Mathematics, Vol. I (Budapest, 1996), Basel: Birkhäuser, 1998, pp. 178–190Google Scholar
  15. 15.
    Eliasson, L.H.: Reducibility and point spectrum for linear quasi-periodic skew-products. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, (electronic), 1998, pp. 779–787Google Scholar
  16. 16.
    Eliasson, L.H.: On the discrete one-dimensional quasi-periodic Schrödinger equation and other smooth quasi-periodic skew products. In: Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995), Volume 533 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Dordrecht: Kluwer Acad. Publ., 1999, pp. 55–61Google Scholar
  17. 17.
    Eliasson, L.H.: Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Herman, M.R.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), (1983)Google Scholar
  19. 19.
    Ince, E.L.: Ordinary Differential Equations. New York: Dover Publications, 1944Google Scholar
  20. 20.
    Jitomirskaya, S.Y.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)Google Scholar
  21. 21.
    Johnson, R.: The recurrent Hill’s equation. J. Diff. Eq. 46, 165–193 (1982)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger equation. J. Diff. Eq. 91, 88–110 (1991)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438, (1982)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Johnson, R.A.: A review of recent work on almost periodic differential and difference operators. Acta Appl. Math. 1(3), 241–261 (1983)zbMATHGoogle Scholar
  25. 25.
    Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on T × S L(2, R). PreprintGoogle Scholar
  26. 26.
    Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164(2), 421–432 (1994)zbMATHGoogle Scholar
  27. 27.
    Last, Y.: Almost everything about the almost Mathieu operator. I. In: XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge MA: Internat. Press, 1995, pp. 366–372Google Scholar
  28. 28.
    Moser, J.: An example of schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59, 39–85 (1984)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Puig, J.: Reducibility of linear differential equations with quasi-periodic coefficients: A survey. Barcelona: Preprint University of Barcelona, 2002, Available at http://www.maia. ub.es/∼puig/preprints/qpred.psGoogle Scholar
  31. 31.
    Puig, J, Simó, C.: Analytic families of reducible linear quasi-periodic equations. In progress 2003Google Scholar
  32. 32.
    Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math. 3(4), 463–490 (1982)zbMATHGoogle Scholar
  33. 33.
    Simon, B.: Schrödinger operators in the twenty-first century. In: Mathematical physics 2000, London: Imp. Coll. Press, 2000, pp. 283–288Google Scholar
  34. 34.
    Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Statist. Phys. 46(5-6), 861–909 (1987)Google Scholar
  35. 35.
    van Mouche, P.: The coexistence problem for the discrete Mathieu operator. Commun. Math. Phys. 122(1), 23–33 (1989)zbMATHGoogle Scholar
  36. 36.
    Yakubovich, V.A., Starzhinskii V.M.: Linear differential equations with periodic coefficients. 1, 2. New York-Toronto, Ont: Halsted Press [John Wiley & Sons], 1975Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Dept. de Matemàtica Aplicada i AnàlisiUniv. de BarcelonaBarcelonaSpain

Personalised recommendations